A beam with a moment of inertia I and with Young's modulus E will have a
bending stress f at a distance from the Neutral Axis (NA) y and the NA will bend
to a radius R ...in accordance with the following formula.

M / I = s / y = E / R

Important note
W and w as used below for beam concentrated load, total load and uniform distributed load
are assumed to be in units of force i.e. Newtons If they are provided in units of
weight i.e kg then they should be converted into units of force by mutliplying by the gravity constant
g (9.81)

Simply Supported Beam . Concentrated Load

Simply Supported Beam . Uniformly Distributed Load

Cantilever . Concentrated Load

Cantilever . Uniformly Distributed Load

Fixed Beam . Concentrated Load

Fixed Beam . Uniformly Distributed Load

Torsion /Shear

Poisson's Ratio = ν = (lateral strain / primary strain )

Shear Modulus G = Shear Stress /Shear Strain

G = τ / ε = E / (2 .( 1 + ν ))

General Formula for Torsion

A shaft subject to a torque T having a polar moment of inertia J
and a shear Modulus G will have a shear stress q at a radius r and an
angular deflection θ over a length L as calculated
from the following formula.

T / J = G . θ / L = t / r

More detailed notes on torsion calculations are found at webpage Torsion

Consider a cylinder with and internal diameter d 1, subject to an
internal pressure p 1. The external diameter is d 2 which
is subject to an external pressure p 2.
The radial pressures at the surfaces are the same as the applied pressures therefore

σr = A + B / r 2σt = A - B / r 2

The radial pressures at the surfaces are the same as the applied pressures therefore

- p1 = A + B / r 12
-p2 = A + B / r 22

The resulting general equations are known as Lame's Euqations and are shown as follows